I learned the same thing 60 years ago but I never really understood because it was never explained to me until my wife became a language therapist and I remembered to ask her about this. (no more that 10 years ago)
There are rarely used words that are taken from other languages that might fit an explanation, but if you are talking about common everyday words used in English, "w" is not a vowel by itself. It is a vowel when used in some diphthongs like draw, cow or few. In these words you couldn't get that vowel sound unless you put the "w" on there so in those cases it is a vowel.
Check out this earlier thread.
aeiou and sometimes y and w
I completely agree with Robert on this. I think that there are a few letters that uniquely shade adjacent vowels, and that we would never consider vowels in themselves. Both L and R are good examples:
R and L are vocalic (vowels) in several languages, including English. In some languages, it is "official" in the orthography ( Czech: prst "finger"; zblbl "I made a mistake") In American English we often pronounce them as vocalic in unstressed final position: bottle; butter.
In my opinion, that vowel shading test for vowelhood springs up to explain the inclusion of W in the axiom, but it doesn't hold Wtr.